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The owner of a chain of clothing stores is comparing the monthly profit earned in the past year from four different store locations. She calculated the mean and standard deviation of the monthly profit, in dollars, for each location, as shown in the table.

\begin{tabular}{|c|c|c|c|}
\hline
Location [tex]$A$[/tex] & Location B & Location C & Location D \\
\hline
Mean [tex]$=23,124.70$[/tex] & Mean [tex]$=24,842.18$[/tex] & Mean [tex]$=20,833.33$[/tex] & Mean [tex]$=21,432.82$[/tex] \\
[tex]$SD =1,553.43$[/tex] & [tex]$SD =1,617.20$[/tex] & [tex]$SD =1,462.15$[/tex] & [tex]$SD =1,512.10$[/tex] \\
\hline
\end{tabular}

For which store location does [tex]$68\%$[/tex] of the data lie between [tex]$\$[/tex]19,371.18[tex]$ and $[/tex]\[tex]$22,295.48$[/tex]?

A. Location D
B. Location A
C. Location C
D. Location B


Sagot :

To determine for which store location 68% of the data falls between \[tex]$19,371.18 and \$[/tex]22,295.48, we will calculate the interval for each location using the mean and the standard deviation (SD). This interval is given by mean ± 1 SD.

1. Location A:
- Mean: \[tex]$23,124.70 - SD: \$[/tex]1,553.43
- Interval: [tex]\(23,124.70 - 1,553.43\)[/tex] to [tex]\(23,124.70 + 1,553.43\)[/tex]
[tex]\[ = 21,571.27 \text{ to } 24,678.13 \][/tex]

2. Location B:
- Mean: \[tex]$24,842.18 - SD: \$[/tex]1,617.20
- Interval: [tex]\(24,842.18 - 1,617.20\)[/tex] to [tex]\(24,842.18 + 1,617.20\)[/tex]
[tex]\[ = 23,224.98 \text{ to } 26,459.38 \][/tex]

3. Location C:
- Mean: \[tex]$20,833.33 - SD: \$[/tex]1,462.15
- Interval: [tex]\(20,833.33 - 1,462.15\)[/tex] to [tex]\(20,833.33 + 1,462.15\)[/tex]
[tex]\[ = 19,371.18 \text{ to } 22,295.48 \][/tex]

4. Location D:
- Mean: \[tex]$21,432.82 - SD: \$[/tex]1,512.10
- Interval: [tex]\(21,432.82 - 1,512.10\)[/tex] to [tex]\(21,432.82 + 1,512.10\)[/tex]
[tex]\[ = 19,920.72 \text{ to } 22,944.92 \][/tex]

Now, we examine if the interval between \[tex]$19,371.18 and \$[/tex]22,295.48 lies within the calculated intervals for each location.

- Location A: The interval is \[tex]$21,571.27 to \$[/tex]24,678.13. The given range does not fall within this interval.
- Location B: The interval is \[tex]$23,224.98 to \$[/tex]26,459.38. The given range does not fall within this interval.
- Location C: The interval is \[tex]$19,371.18 to \$[/tex]22,295.48. The given range exactly matches this interval.
- Location D: The interval is \[tex]$19,920.72 to \$[/tex]22,944.92. The given range does not fall within this interval.

Therefore, the correct answer is:

C. Location C
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